On a boundary value problem for conically deformed thin elastic sheets

We consider a thin elastic sheet in the shape of a disk that is clamped at its boundary such that the displacement and the deformation gradient coincide with a conical deformation with no stretching there. We define the free elastic energy as a variation of the von K'arm'an energy, that penalizes bending energy in $L^p$ with $p\in (2,\frac{8}{3})$ (instead of, as usual, $p=2$). We prove ansatz free upper and lower bounds for the elastic energy that scale like $h^{p\mathrm{/}(p-1)}$, where $h$ is the thickness of the sheet.